page 1 x-ray crystallography: "molecular photography" object irradiate scattering lens...

X-ray crystallography: "molecular photography"

Object

Irradiate

Scattering

lens

Combination

Image

• Need wavelengths smaller than or on the order of the size of the objects that we wish to see - for atomic spacings, need ~1 Å wavelength. ==> X-rays

• X-rays: two fundamental problems:

1. No X-ray lens! But: we know the mathematical relationship between scattering and the image (Fourier transform) - computer can simulate the lens

2. Scattering from one molecule is very weak

==> crystals: amplify scattering by aligning many (~1012) molecules

X-ray scattering

• A charge which interacts with an electromagnetic wave experiences a force and oscillates.

• Oscillating (i.e. accelerating) charges act as new sources.

• Strength of scattering 1/mass, so electrons scatter >> protons

• Strength of scattering no. of electrons, so C, N, O, S, P scatter >> H

• Scattering from more than one point leads to interference between the scattered radiation

a = amplitudeh = spatial frequency (how many repeats/unit length)= relative separation

1 cycle = 360° = 2 radWave travels in one cycleAt point x, travel 2x/ part of cycle

y1=a cos (2hx)y2=a cos (2hx+ )

-a

a

x

y

= phase angle = 2

A wave with arbitrary phase can be represented by y = a [cos (2hx) + i sin (2hx)] = a ei2hx

Waves as vectors in an Argand diagram

|F|(cos + i sin )=|F|ei

Real

Imaginary

0°180°

270°

90°

F

A = |F|cos

B = |F|sin

|F| = (A2 + B2)1/2

= tan-1 (B/A)Note: the sign of both A and B determines the quadrant of

Two Point Scatterers

2

#1

#2

E

B

Incident radiationis in phase

Radiation scatteredfrom two pointsat angle 2will in general be out of phase

Observe at distance >> spacing of scatterers

2

#1 (origin)

#2

x12

l1

l2

x12

ni

ni = unit vector in direction of incident radiation ns

ns = unit vector in direction of scattered radiation

Define unit vectors:

Path length difference l l1 l2 ˆ n s x ˆ n i x ˆ n s ˆ n i x

Phase shift of radiation scattered from #2 relative to that scattered from #1:

2 l

2

ˆ n s ˆ n i x

ˆ n s

ˆ n i

s=ˆ n s ˆ n i

Scattering Vector

Ewald sphereradius = 1/

We define the scattering vector s as shown. The tip of s lies on the"Ewald sphere" (this is also called the"sphere of reflection").

Since the scattering angle is 2, the magnitude of the scatteringvector is:

s 2sin( )

In terms of s, the phase shift of radiationscattered from #2, relative to thatscattered from #1, is:

ns

ni

s

x12

2

ˆ n s ˆ n i x

2sx

2

ˆ n s ˆ n i x

2sx

For each scatterer, only a fraction of the incident radiation will be scattered; the ratio of scattered to incident radiation is called the "scattering factor" or "scattering power", denoted f.

The contribution of each scatterer to the total scattering amplitude of a group will then include both the scattering factor f (the magnitude of its contribution) and a phase relative to an origin.

E0

f E0

incident scattered

contribution = f ei() length=f

Re

Im

Multiple scatterers - at a given angle 2, see sum of all scattering in that direction

For a group of scatterers, the total scattered wave at a given observation angle is given by the sum of all contributions:

x2

x3

1

2

3

x1origin

F(s) f j

all scatterers j e

i(j) fj

all scatterers j ei(2 sx j )

F(s) = | F(s)| ei is called the structure factor

|F(s)| is the amplitude and is the phase of the wave that results from superposition of the scattered waves of the individual atoms in the direction given by s (|s| = 2 sin /).

Real axis

Im. a

xis

f2

f3

f1

F

For electromagnetic radiation, the observed intensity I(s) is given by the absolute square of the amplitude I(s) = F*(s) F(s).

Scattering pattern = Fourier transform of the object

The structure factors are the component waves that represent the object. Structure factors at higher angle are higher frequency components, provide more detail.

Object Diffraction pattern Image Fourier transform Fourier transform

Any complex waveform can be built up by superposition of waves of different frequencies(Fourier synthesis), or decomposed into its component frequencies (Fourier analysis).

Crystal = motif that repeats in spaceThe repeat unit = UNIT CELL

Diffraction from crystals

The repeat lengths define a lattice

c

b

a

The crystal can be considered as a single motif that is laid down on every lattice point.

Scattering from a 1-D lattice of spacing a

a.…. …..

-4 0-3 -2 -1 1 2 3 4n =

There are 2N + 1 identical scatterers. The scattering from the nth scatterer at s is f(s) ei2ns a

N sin[(2N+1) s a]The total scattering from the array is F(s) = f(s) ei2n s a = f(s)

n = -N sin( s a)

We observe intensity, so for N scatterers the intensity profile is given by

sin2[(2N+1) s a] I(s) sin2( s a)

sin2[(2N+1) s a]I(s)

sin2( s a)

For large N, I(s) is onlysignificant when the denominator is 0, i.e.,

s a = h, h is an integer

This is known as the Laue condition.

We see diffracted intensity from a lattice of spacing a only when s a = h (projection of s on a). |s| = h / |a| , i.e. see scattering in only in discrete planes perpendicular to a, spaced at integral multiples of 1/a.

s lies on planes perpendicular to a, spaced at 1/a s lies on Ewald sphere

We observe intensity only when the reciprocal lattice intersectsthe Ewald sphere (meets both requirements for s)

3-d:s a = h s b = k s c = l

Define reciprocal lattice:1. r.l. axis length inversely related to corresponding real axis

a a* = 1 b b* = 1 c c* = 1

2. A r.l. axis is perpendicular to the other two corresponding real axes, e.g., a* is perpendicular to the b, c plane.

3. The distance from the origin to any r.l. point is given by

d* = ha* + kb* + lc*

Laue equations

Tip of s at lattice points

Scattering Vectorˆ n s

ˆ n i

s=ˆ n s ˆ n i

radius = 1/

ˆ n s

ˆ n i

radius = 1/

2sin()

nd * Where d* is a reciprocal lattice spacing

=> see scattering where the reciprocal lattice intersects the Ewald sphere

General scattering: s lies on Ewald sphere Single crystal: s defines reciprocal lattice

d*

|s| =

Bragg's treatment of diffraction as reflection from planes in the crystal lattice

Scattered waves from parallel planes are in phase and constructively interfere if their path lengths differ by an integral number of wavelengths:

2 d sinn

With d the spacing between planes.

Planes denoted (h,k,l) - integers indicate the number of times that the planes intersect the unit cell edge. The planes contain the scattering centers.

Planes populated with many scatterers give strong scattering.

Finer spacing of planes (larger h, k, l) corresponds to more interatomic detail - higher resolution

2sin()

nd * Where d* is a reciprocal lattice spacing|s| =

Rearrange Bragg's Law:2sin()/ =nd

d = spacing of lattice planes

ˆ n s

ˆ n i

radius = 1/

2d*ˆ n s

ˆ n i

radius = 1/

d*n = 0

n = 2

n = 3

n = 1

d* = 1/d

d d/2

= spacing ofr.l. points

The reciprocal lattice points indexed by h, k, l are equivalent to reflection from crystal planes (h k l) in the Bragg treatment, hence they are often called "reflections".

There is an inverse relationship between the distances of the real space lattice and the spacing of the reciprocal lattice points.

Single molecule would give a continuous scattering pattern

Incidentradiation

Scatteredradiation

Intensity

Scatterin

g angle

Intensity

Scatterin

g angle

Scatteredradiation

Incidentradiation

A lattice of identical points acts as a diffraction grating

When a copy of the molecule is placed on each point of the lattice, its continuous transform is multiplied by the lattice transform. This gives a diffraction pattern in which the spacing of spots is determined by the lattice spacing and the intensity is determined by scattering profile of the molecule.

Motifs and latticesCrystal

* =

The motif, or unit cell, may be formed from more than one molecule. If there is more than one molecule in the unit cell, they may be related by a symmetry operation that applies to every point in the crystal; this is called "crystallographic symmetry".

A

A’

B

B’ The two bird motif can generate the rest of the crystal by translations, and is called the “unit cell”

One bird, by application of the crystallographic symmetry and the translations can generate the whole crystal, and so is called the “asymmetric unit”

The asymmetric unit is the set of molecules that, when transformed according to the crystallographic symmetry, generates the unit cell. The asymmetric unit can be more than one molecule; the relationships among multiple molecules in the asymmetric unit is "noncrystallographic" or "local" symmetry.

Lattices and space groups• Molecules in the unit cell can be related by various kinds

of symmetry operations. The set of symmetry operations that generate the unit cell from the asymmetric unit is called the space group.

• Enantiomorphic molecules have restricted types of symmetry:

– Rotations

• Pure

- Centering operations

• Screw180 180

P C I F

[P]

[P,C][P,C,I,F]

[P,I]

[P][P]

[P,I,F]

Particular types of crystallographic symmetry impose particular constraints on the geometry of the crystal lattice, giving rise to 7 “crystal systems”.

Orthorhombic symmetry generated by perpendicular twofold axes.e.g. twofolds about a single point: 222 symmetry

180°

180°

Angles between axes are constrained by symmetry to be 90°.

The three perpendicular twofolds constrain the orthorhombic cell to have 90° angles

Hence,a ≠ b ≠ c but = = = 90°

a

b

c

Data collection: move crystal so that its associated reciprocal lattice intersects the Ewald sphere.

1. The reciprocal lattice obeys symmetry relationships of the corresponding real lattice.

ˆ n s

ˆ n i

detector

X-ray beam

2. The symmetry of the unit cell contents is present in the scattering pattern. Therefore, it is generally not necessary to measure diffraction data in all scattering directions, as some reflections are equivalent to others by symmetry.

Scattering pattern = Fourier transform of the object

The structure factors are the component waves that represent the object. Structure factors at higher angle are higher frequency components, provide more detail.

Object Diffraction pattern Image Fourier transform Fourier transform

Any complex waveform can be built up by superposition of waves of different frequencies(Fourier synthesis), or decomposed into its component frequencies (Fourier analysis).

To reconstruct the image, we need the component structure factors. However , the observed intensity I(s) = F*(s) F(s). When we measure the diffraction pattern, we measure I(s), so we only obtain the amplitude of the structure factor, and lose its phase. This is the central problem in crystallography.

Atomic scattering

• A charge which interacts with an electromagnetic wave

experiences a force and oscillates.

• Oscillating (i.e. accelerating) charges act as new sources.

• The atomic scattering factor f describes the scattering given the

arrangement of electrons in the atom (interference

between charges)

• For free electrons, the scattered radiation has a constant phase

shift of 180° relative to the incident radiation

Anomalous scattering

Scattering by free electrons: energy (wavelength) invariant; constant phase shift. Real atoms: electrons are not free, but bound in orbitals

-- absorption occurs near energies corresponding to transitions between orbitals-- the scattered radiation experiences a phase shift that is energy dependent

Energy ( = hc/) Scattering angle

f

"Normal" atomic scattering Anomalous scattering

f

The atomic scattering factor f can be described by the normal (energy-invariant, treating electrons as free scatterers) and anomalous (energy dependent) scattering. Because of the energy-dependent phase shift, the scattering must now be described as a complex number. The real part of the anomalous scattering is f' , and the imaginary component is f''. Away from the absorption edge, these anomalous "corrections" are generally small, but near the edge they are significant.

Real axis

Im. a

xis

f'fnormal

f''f

Atom at origin Atom at general position in unit cell

Real axis

Im. a

xis

f'

fnormal

f''f

page 1 x-ray crystallography: "molecular photography" object irradiate scattering lens...

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